Q22.    Express 103(97) as a difference of squares.

A22.    103(97) = (100 + 3)(100 - 3)

                        = 1002 - 32 = 9991

 

Q23.    The shaded area between a circle of radius r and an ellipse is πr2/2. Find x.

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A23.    Shaded area = πr2/2 = πr2 - πab, where a = x and b =r.

                                    πr2/2 = πr2 - πxr

                                    r/2 = r - x

                                    x = r/2.

 

Q24.    What is the perimeter of the triangle defined by A( 0,0 ) , B( 4,0 ) and C( 4,4 ) ?

A24.    Perimeter        = AB + BC + CA

                                    = 4 + 4 + 4√2 = 8 + 4√2

 

 

 

Q25.    A cone-shaped cup and a cylinder have the same height and the same diameter. How many 

            cups of water from the cone are needed to fill the cylinder ?

A25.    3

 

Q26.    The Pythagorean triple that includes 11 also includes two consecutive integers. Find them.

A26.    112 + x2 = (x+1)2

            121 + x2 = x2 + 2x +1

            x = 60

            The full triple is 11, 60 and 61.

 

Q27.    Find the distance between the plane and the horizon:

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A27.    d2 + 61002 = 61102.

            d = 349 km.

 

 

 

Q28.    The cost of two pizzas of the same thickness is proportional to area. A pizza 10 cm in             diameter costs $5. How much should a 20 cm one cost ?

A28.    (20/10)2($ 5) = $ 20.

 

Q29.    Factor a3 + b3 + c3 -3abc

A29.    (a + b + c)(a2 + b2 + c2- ab -ac -bc)

 

Q30.    Why can't a Pythagorean triple include 3 odd numbers ?

A30.    The square of any odd number is odd, and the sum of any two odd numbers is even. Since             the square root of an even number is also even, then c = √(a2 +b2) would not be odd.

 

Q31.    Find the next number in the sequence

            2, 12, 36, 80, 150 , ...

A31.    2 = 12 + 13

            12 = 22 + 23

            36 = 32 + 33

            sixth term = 62 + 63 = 252.

 

Q32.    One way of testing whether 10277 is prime is to test its divisibility by all primes < y.

            What should y be ?

A32.    √10277 = 101.4. Incidentally 10277 = 43* 239.

 

 

 

Q33.    How many real roots does the equation x4 + x3 + x2 + x = 0 have ?

A33.    Just two.         x4 + x3 + x2 + x = 0.

                                    x3( x + 1) + x (x + 1) = 0.

                                    (x + 1 )( x3 + x) = 0.

                                    x(x+1)(x2+1) = 0.

                                    so x = 0 or -1.

 

Q34.    Express the area of the triangle in terms of angles A and B and its base d.

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A34.

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(continued)

 

 

                                    xtanA = tanB (d - x)

                                    xtanA = dtanB - xtanB

                                    x(tanA + tanB) = dtanB











Q35.    When reflected in y = x, the equation of a certain line remains unchanged. If this line             passes through (2 , 2), what is its equation ?

 

A35.    For the equation of a line to be unaffected after such a reflection, its slope must be -1.

                                                y = mx + b

                                                2 = -1(2) + b

                                                b = 4

                                                y = -x + 4

 

Q36.    When expanded, ( 3x + 2 )6 has seven terms. What is the coefficient of the seventh term ?

A36.    Using Pascal's triangle:

                                                            1

                                                1                      1

                                    1                      2                      1

                     1               3                      3                    1

           1                        4           6                        4                       1

     1       5                10                      10                    5                      1

1 6           15                     20                    15                    6                      1


The above line applies to a power of 6 so that:

            (3x)6 + 6(3x)5(2) + 15(3x)4(22) + 20(3x)3(23) + 15(3x)2(24) + 6(3x)(25) + 26



            The middle term's coefficient is 4320.



Q37.    Why are sewer caps round and not square-shaped?

 

A37.                The round cover sits on a lip that is smaller than the cover. Because its diameter is             constant, the cover cannot fall through. Keep in mind that the diagonal of a square is             about 1.41 times as long as one of its sides. As a result if the side of a square-

            cover was lined up with the diagonal of the hole, it would slide through in spite of its lip.